DARUAN: Data Re-Uploading Activation

• DARUAN is a quantum-inspired architecture that uses repeated data encoding interleaved with trainable gates to achieve universal function approximation.
• The framework supports single-qubit, photonic, and bosonic implementations, leveraging minimal circuit settings for expressive, resource-efficient activations.
• Empirical results indicate high classification accuracy, improved trainability, and enhanced noise resilience compared to traditional quantum and classical models.

Data Re-Uploading ARtive Activation (DARUAN) is a quantum and quantum-inspired architectural paradigm that realizes highly expressive, resource-efficient function approximation and nonlinear activation by interleaving repeated data encoding with parameterized operations in minimal quantum (or classical single-qubit) settings. Originating from the need to surmount expressivity limits and training bottlenecks in variational quantum circuits (VQCs) and feedforward architectures, DARUAN engineering is now found across quantum classification, reinforcement learning, photonic and bosonic implementations, and quantum-inspired neural networks (notably Kolmogorov-Arnold Networks, KANs).

1. Core Principle and Universal Approximation via Data Re-Uploading

DARUAN is fundamentally a circuit design principle in which classical or quantum data is injected (encoded) into a small quantum register repeatedly at multiple positions in a variational circuit—between all layers of trainable unitaries—instead of only once at the beginning. Each encoding operation, or "re-uploading," is interleaved with parameterized gates, typically SU(2) rotations or more general unitary maps. Formally, in the single-qubit classical-input case, the circuit is

$$|\psi_\mathrm{out}(x;\theta)\rangle = U_L(\theta_L)\, E(x) \cdots U_1(\theta_1)\, E(x)\, |0\rangle,$$

where $E(x)$ denotes a data-encoding gate such as a Pauli rotation, and $U_k(\theta_k)$ is a variational layer.

This architecture leads to a functional output with a truncated Fourier (or polynomial, for quantum inputs) spectrum whose richness grows exponentially in both the number of re-upload layers and the number of distinct data weights, enabling universal approximation theorems for continuous functions on both classical and quantum domains (Pérez-Salinas et al., 2019, Mauser et al., 7 Jul 2025, Cha et al., 23 Sep 2025). In the quantum-input generalization, an ancilla qubit interacts sequentially with multiple fresh copies of an input quantum state via entangling unitaries, with the information transfer formalized as a cascade of completely positive and trace-preserving (CPTP) maps.

2. Circuit Implementations and Activation Mechanisms

The architectural blueprint of DARUAN spans single-qubit, bosonic, and multi-qubit realizations:

• Single-qubit repetitive encoding: Each layer alternates between data-encoding rotations and trainable SU(2) gates, yielding activation functions that are finite, tunable Fourier or trigonometric series in the input (Pérez-Salinas et al., 2019, Mauser et al., 7 Jul 2025).
• Quantum input processing: For density matrix inputs $\rho_\mathrm{in}$, DARUAN acts on an ancilla and data register; in each layer, a controlled-unitary $U_k(\theta_k)$ entangles the ancilla with a fresh copy of $\rho_\mathrm{in}$, followed by resetting the data register. The system output is a function of the ancilla expectation values after L rounds, with nonlinearity realized by entanglement, measurement, and the element-wise scaling of the ancilla’s Bloch vector by the quantum data coordinates (Cha et al., 23 Sep 2025).
• Photonic and bosonic circuits: Photonic implementations use Mach–Zehnder Interferometers to realize encoding and optimization operations, often in dual-rail encoding (single-photon subspace), and generalize to bosonic multi-photon spaces where SU(2) beamsplitter layers alternate with programmatic phase shifts, with classical inputs modulating the phase shifters (Mauser et al., 7 Jul 2025, Ono et al., 2022).

The logic of DARUAN activation is to use the circuit (ancilla’s post-L-layer measured observable) as a highly expressive, trainable, and smooth function of the data input, analogous to classical learnable nonlinear activations, but with richer frequency content and tunable spectral density (Jiang et al., 17 Sep 2025, Hsu et al., 4 Dec 2025).

3. Architectural Extensions in Neural and Sequential Models

DARUAN modules have been incorporated as learnable activation functions—Quantum Variational Activation Functions (QVAFs)—in quantum-inspired neural architectures:

• KANs and QKANs: DARUAN is embedded as the univariate activation function on each edge of a Kolmogorov-Arnold Network, forming quantum-inspired KAN (QKAN) models. Each univariate activation $f(x;\theta)$ is the measured expectation of a single-qubit data re-uploading circuit with adjustable data weights, providing exponential parameter efficiency for a given spectral accuracy compared to classical Fourier or spline activations (Jiang et al., 17 Sep 2025, Sharma et al., 9 Oct 2025).
• RNNs and LSTMs: The QKAN-LSTM replaces the affine transforms inside each LSTM gate with sums of DARUAN modules, exploiting the exponentially enriched frequency spectrum of the activation (number of unique harmonics up to $2^L$), directly improving expressivity while greatly reducing parameter count (Hsu et al., 4 Dec 2025).
• Deep quantum neural blocks: In full quantum feedforward architectures, each neuron can be instantiated as a DARUAN block (single ancilla + L re-upload layers), with inter-layer feature mixing realized via parameterized unitaries among ancillas. The full network inherits universal approximation guarantees for continuous functionals over quantum states (Cha et al., 23 Sep 2025).

A table summarizing modular uses follows:

DARUAN Context	Circuital Role	Expressivity Source
Quantum classifier (single qbit)	Alternating data/gate layers	Truncated Fourier series
QKAN/QKAN-LSTM	QVAF on each edge/gate	Exponential frequency spectrum
Quantum-input functional	Entangling ancilla/data block	Polynomials in input coordinates

4. Empirical Results, Trainability, and Noise Robustness

DARUAN circuits consistently demonstrate improved performance and training behavior across experimental platforms:

• Classification: Photonic one-qubit experiments achieve >90% accuracy on toy 2D/9D/20D tasks, matching or exceeding classical discriminant analysis and SVM baselines (Mauser et al., 7 Jul 2025). Bosonic DARUAN-classifiers (two-mode, two-photon) achieve ≈94% correct rate on nontrivial geometric tasks (Ono et al., 2022).
• Reinforcement learning: DARUAN-augmented VQC DQNs solve CartPole and Acrobot environments up to 2× faster and more reliably than baseline VQCs. Gradient norms and variances remain large or grow with increasing qubit number, even as the circuit approaches a 2-design, thus avoiding barren plateaus due to continual injection of fresh data and RL target non-stationarity (Coelho et al., 21 Jan 2024).
• Trainability: Parameter-shift-based gradients permit efficient optimization, with single-qubit circuits exhibiting smooth loss landscapes and fast convergence. VC-dimension analyses yield finite, polynomial sample-complexity scaling, supporting generalizability (Mauser et al., 7 Jul 2025).
• Noise resilience: Pulse-native DARUAN implementations, in which the data encoding and variational evolution are mapped directly to trainable hardware control pulses (e.g., on superconducting transmons), show marked improvements in fidelity and generalization under realistic amplitude, phase, and depolarizing errors compared to gate-based circuits (Acedo et al., 11 Dec 2025).

In all tested settings, DARUAN uniquely mitigates parameter inefficiency, spectral inflexibility, and trainability limits relative to both classical and standard quantum circuits.

5. Function Approximation Rate and Parameter Efficiency

DARUAN’s efficiency is quantifiable via its function-approximation spectrum:

• Fourier expansion: With r re-uploading layers, the activation spectrum supports up to $3^r - 1$ unique frequencies (using distinct weights or geometrically spaced weights), enabling C^m-norm error $O(2^{-r(k+1-m)})$ for $C^{k+1}$ targets—exponentially faster than the $O(\varepsilon^{-1/(k+1-m)})$ scaling of classical Fourier networks (Jiang et al., 17 Sep 2025).
• Comparison with splines: For the same frequency content, a KAN with B-spline activations requires O(2L) parameters per unit; DARUAN circuits realize equivalent expressivity with ≈25–50% fewer parameters for moderate L across regression and classification benchmarks (Sharma et al., 9 Oct 2025).
• Scalability mechanisms: Layer extension (incrementally growing re-upload depth) and bottlenecked architectures (as in HQKAN and HQKAN-LSTM) allow deep/wide networks to retain manageable parameter counts while maintaining or improving task accuracy (Jiang et al., 17 Sep 2025, Hsu et al., 4 Dec 2025).

6. Physical Implementations and Hardware Considerations

• Photonic and bosonic chips: Dual-rail photonic chips with Mach–Zehnder Interferometers and single/dual photon input states realize all elements of DARUAN circuits, including full experimental trainability validation and explicit resource analysis (gate counts, extinction, thermal cross-talk) (Mauser et al., 7 Jul 2025, Ono et al., 2022).
• Superconducting transmon platforms: Small-scale DARUAN ansätze implemented in real quantum processors achieve robust optimization and classification under practical timing and hardware constraints, albeit with remaining bottlenecks in classical-to-quantum data transfer and inference speed (Tolstobrov et al., 2023).
• Pulse-level control: For NISQ hardware, pulse-native DARUAN modules reduce total execution time, improve tolerance to pulse errors, and smooth the training landscape, with practical training requiring efficient pulse-programming and feedback loops (Acedo et al., 11 Dec 2025).

7. Outlook: Open Problems and Future Directions

DARUAN formalism has established a universal, parameter-efficient, and noise-robust mechanism for nonlinear function approximation and neural activation in quantum and quantum-inspired learning. Key open questions include:

• Scaling behavior on high-dimensional, continuous-action, or generative tasks; interplay with actor–critic RL and unsupervised modeling (Coelho et al., 21 Jan 2024).
• Analytical understanding of gradient variance and effective dimension under deep re-uploading and physical noise (Jiang et al., 17 Sep 2025, Acedo et al., 11 Dec 2025).
• Integration of pulse-based and gate-based controls, and extension to genuine deep quantum feature extraction (multi-block DARUAN networks operating directly on quantum intermediate outputs) (Cha et al., 23 Sep 2025).
• Further hardware advances in photonic, bosonic, and pulse-programmed superconducting architectures to fully exploit DARUAN’s expressivity and trainability.

DARUAN, as a quantum variational activation paradigm, provides a foundational building block for quantum and quantum-inspired models with applications across advanced function regression, vision, language modeling, time series, and real-world control (Cha et al., 23 Sep 2025, Mauser et al., 7 Jul 2025, Jiang et al., 17 Sep 2025, Hsu et al., 4 Dec 2025, Acedo et al., 11 Dec 2025).